Theorem lnolin 8415

Description: Basic linearity property of a linear operator.

Hypotheses

Ref	Expression
lnoval.1	|- X = (Base` U)
lnoval.2	|- Y = (Base` W)
lnoval.3	|- G = (+v` U)
lnoval.4	|- H = (+v` W)
lnoval.5	|- R = (.s` U)
lnoval.6	|- S = (.s` W)
lnoval.7	|- L = (U LnOp W)

Assertion

Ref	Expression
lnolin	|- (((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) /\ (A e. X /\ B e. CC /\ C e. X)) -> (T` (AG(BRC))) = ((T` A)H(BS(T` C))))

Proof of Theorem lnolin

Step	Hyp	Ref	Expression
1		opreq1 3968	. . . . . 6 |- (u = A -> (uG(vRw)) = (AG(vRw)))
2	1	fveq2d 3728	. . . . 5 |- (u = A -> (T` (uG(vRw))) = (T` (AG(vRw))))
3		fveq2 3724	. . . . . 6 |- (u = A -> (T` u) = (T` A))
4	3	opreq1d 3975	. . . . 5 |- (u = A -> ((T` u)H(vS(T` w))) = ((T` A)H(vS(T` w))))
5	2, 4	eqeq12d 1489	. . . 4 |- (u = A -> ((T` (uG(vRw))) = ((T` u)H(vS(T` w))) <-> (T` (AG(vRw))) = ((T` A)H(vS(T` w)))))
6		opreq1 3968	. . . . . . 7 |- (v = B -> (vRw) = (BRw))
7	6	opreq2d 3976	. . . . . 6 |- (v = B -> (AG(vRw)) = (AG(BRw)))
8	7	fveq2d 3728	. . . . 5 |- (v = B -> (T` (AG(vRw))) = (T` (AG(BRw))))
9		opreq1 3968	. . . . . 6 |- (v = B -> (vS(T` w)) = (BS(T` w)))
10	9	opreq2d 3976	. . . . 5 |- (v = B -> ((T` A)H(vS(T` w))) = ((T` A)H(BS(T` w))))
11	8, 10	eqeq12d 1489	. . . 4 |- (v = B -> ((T` (AG(vRw))) = ((T` A)H(vS(T` w))) <-> (T` (AG(BRw))) = ((T` A)H(BS(T` w)))))
12		opreq2 3969	. . . . . . 7 |- (w = C -> (BRw) = (BRC))
13	12	opreq2d 3976	. . . . . 6 |- (w = C -> (AG(BRw)) = (AG(BRC)))
14	13	fveq2d 3728	. . . . 5 |- (w = C -> (T` (AG(BRw))) = (T` (AG(BRC))))
15		fveq2 3724	. . . . . . 7 |- (w = C -> (T` w) = (T` C))
16	15	opreq2d 3976	. . . . . 6 |- (w = C -> (BS(T` w)) = (BS(T` C)))
17	16	opreq2d 3976	. . . . 5 |- (w = C -> ((T` A)H(BS(T` w))) = ((T` A)H(BS(T` C))))
18	14, 17	eqeq12d 1489	. . . 4 |- (w = C -> ((T` (AG(BRw))) = ((T` A)H(BS(T` w))) <-> (T` (AG(BRC))) = ((T` A)H(BS(T` C)))))
19	5, 11, 18	rcla43v 1882	. . 3 |- ((A e. X /\ B e. CC /\ C e. X) -> (A.u e. X A.v e. CC A.w e. X (T` (uG(vRw))) = ((T` u)H(vS(T` w))) -> (T` (AG(BRC))) = ((T` A)H(BS(T` C)))))
20		lnoval.1	. . . . . 6 |- X = (Base` U)
21		lnoval.2	. . . . . 6 |- Y = (Base` W)
22		lnoval.3	. . . . . 6 |- G = (+v` U)
23		lnoval.4	. . . . . 6 |- H = (+v` W)
24		lnoval.5	. . . . . 6 |- R = (.s` U)
25		lnoval.6	. . . . . 6 |- S = (.s` W)
26		lnoval.7	. . . . . 6 |- L = (U LnOp W)
27	20, 21, 22, 23, 24, 25, 26	islno 8414	. . . . 5 |- ((U e. NrmCVec /\ W e. NrmCVec) -> (T e. L <-> (T:X-->Y /\ A.u e. X A.v e. CC A.w e. X (T` (uG(vRw))) = ((T` u)H(vS(T` w))))))
28	27	pm3.27bda 421	. . . 4 |- (((U e. NrmCVec /\ W e. NrmCVec) /\ T e. L) -> A.u e. X A.v e. CC A.w e. X (T` (uG(vRw))) = ((T` u)H(vS(T` w))))
29	28	3impa 828	. . 3 |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) -> A.u e. X A.v e. CC A.w e. X (T` (uG(vRw))) = ((T` u)H(vS(T` w))))
30	19, 29	syl5 21	. 2 |- ((A e. X /\ B e. CC /\ C e. X) -> ((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) -> (T` (AG(BRC))) = ((T` A)H(BS(T` C)))))
31	30	impcom 351	1 |- (((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) /\ (A e. X /\ B e. CC /\ C e. X)) -> (T` (AG(BRC))) = ((T` A)H(BS(T` C))))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206   LnOp clno 8401

This theorem is referenced by:  lno0 8417  lnocoi 8418  lnoadd 8419  lnosub 8420  lnomul 8421

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-lno 8405

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